Finally, note that in the complementary IR regime, the flow equations receive non-melonic contributions. This is due to the fact that, for a very small cut-off, the sums take values 0 or 1. As pointed in [1]}, the appropriate rescalling is provided by the standard power counting, and the flow equation turn as well to be an autonomous system, which can be solved numerically. However, we have to keep in mind that our model is defined on a compact manifold (\(U(1)^6\)  in the referenced paper, \(U(1)^5\)  in our case). Then, no phase transition can be occurs, and all the non-Gaussian fixed point reached to the Gaussian one when the cut-off tend to 0, except if the radius of the circles tend simultaneously to the infinity.
